Extended optimality in topology design

Most existing studies of 2D problems in structural topology optimization are based on a given (limit on the) volume fraction or some equivalent formulation. The present note looks at simultaneous optimization with respect to both topology and volume fraction, termed here “extended optimality”. It is shown that the optimal volume fraction in such problems — in extreme cases — may be unity or may also tend to zero. The proposed concept is used for explaining certain “quasi-2D” solutions and an extension to 3D systems is also suggested. Finally, the relevance of Voigt’s bound to extended optimality is discussed.

[1]  A. Michell.  LVIII. The limits of economy of material in frame-structures , 1904 .

[2]  Andrej Cherkaev,et al.  Regularization of optimal design problems for bars and plates, part 1 , 1982 .

[3]  R. Kohn,et al.  Optimal design and relaxation of variational problems, III , 1986 .

[4]  George I. N. Rozvany,et al.  Least-weight design of perforated elastic plates. II , 1987 .

[5]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[6]  George I. N. Rozvany,et al.  Structural Design via Optimality Criteria , 1989 .

[7]  G. Rozvany,et al.  Michell layouts for various combinations of line supports. I , 1990 .

[8]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[9]  M. Zhou,et al.  Generalized shape optimization without homogenization , 1992 .

[10]  George I. N. Rozvany,et al.  DCOC: An optimality criteria method for large systems Part I: theory , 1992 .

[11]  C. S. Jog,et al.  Topology design with optimized, self‐adaptive materials , 1994 .

[12]  G. Rozvany,et al.  Extended exact solutions for least-weight truss layouts—Part I: Cantilever with a horizontal axis of symmetry , 1994 .

[13]  Martin P. Bendsøe,et al.  Optimization of Structural Topology, Shape, And Material , 1995 .

[14]  O. Sigmund,et al.  Checkerboard patterns in layout optimization , 1995 .

[15]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[16]  N. Olhoff,et al.  Topology optimization of three-dimensional structures using optimum microstructures , 1998 .

[17]  M. Beckers,et al.  Topology optimization using a dual method with discrete variables , 1999 .

[18]  J. E. Taylor,et al.  A Formulation For Optimal Structural Design with Optimal Materials , 2000 .

[19]  G. Rozvany.  Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics , 2001 .